Statistical mechanics, structure of matter Proposition

Kelvin relations for thermoelectricity

Builds on Thermoelectric effect Onsager reciprocal relations

Statement

Let an isotropic conductor have Seebeck coefficient $S(T)$ , Peltier coefficient $\Pi(T)$ and Thomson coefficient $\mathcal{K}(T)$ at absolute temperature $T$ . Use the conventions of the thermoelectric effect. Assume the linear transport laws and the Onsager reciprocal relations hold. Then

$\Pi = T S$ ;
$\mathcal{K} = T \dfrac{dS}{dT}$ .

Remarks

Write the linear laws with the forces $\nabla(1/T)$ and $E/T$ ; the Onsager symmetry of the cross coefficients is exactly relation 1. Relation 2 follows from energy conservation, which gives $\mathcal{K} = d\Pi/dT - S$ in steady state, together with relation 1. Example: if $S$ is independent of temperature, the Thomson effect vanishes and $\Pi$ grows linearly with $T$ . For a material with $S = 200\,\mu\mathrm{V/K}$ at $T = 300$ K, relation 1 gives $\Pi = 60$ mV, so a current of $1$ A carries $60$ mW of heat. William Thomson, later Lord Kelvin, found these relations before Onsager's theory, by an argument that treated the reversible effects separately from heat conduction.

Going deeper

For a student

These relations were among the first tests of the Onsager reciprocal relations, and they hold well in experiments. In practice, $S(T)$ is measured, and $\Pi$ and $\mathcal{K}$ are computed from it. The figure of merit of a thermoelectric material, $ZT = \sigma S^2 T / \kappa$ , involves only $S$ for this reason.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Entropy concept
Entropy production concept
Local thermodynamic equilibrium concept
Symmetric matrix concept
Thermoelectric effect concept
Onsager reciprocal relations statement
Kelvin relations for thermoelectricity statement

Tag	`N89246VPKQ`
URI	`m2://N89246VPKQ`
Kind	statement, proposition
Writing	W0 stub
MSC	82C35, 80A17
Status	This copy of the site has no build data. Run `m2 status N89246VPKQ`.
About	Thermoelectric effect
Uses	Thermoelectric effect, Onsager reciprocal relations
Changed	3 November 2026